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Lower Bounds for Data Structures

with Space Close to Maximum

Imply Circuit Lower Bounds

Emanuele Viola

Received February 15, 2019; Revised October 4, 2019; Published December 18, 2019

Abstract: Let f :{0,1}n→ {0,1}mbe a function computable by a circuit with unbounded

fan-in, arbitrary gates, wwires and depth d. With a very simple argument we show that them-query problem corresponding to f has data structures with spaces=n+rand time

(w/r)d, for anyr. As a consequence, in the setting wheresis close toma slight improvement on the state of existing data-structure lower bounds would solve long-standing problems in circuit complexity. We also use this connection to obtain a data structure for error-correcting codes which nearly matches the 2007 lower bound by Gál and Miltersen. This data structure can also be made dynamic. Finally we give a problem that requires at least 3 bit probes for

m=nO(1)and evens=m/2−1.

Independent work by Dvir, Golovnev, and Weinstein (2018) and by Corrigan-Gibbs and Kogan (2018) give incomparable connections between data-structure and other types of lower bounds.

ACM Classification:F.2.3

AMS Classification:68Q17

Key words and phrases: data structure, lower bound, circuit, arbitrary gates, wires, error-correcting codes

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1

Introduction

Proving data-structure lower bounds is a fundamental research agenda to which much work has been devoted, see for example [21] and the 29 references there. A static data structure for a function

f :{0,1}n→ {0,1}mis specified by an arbitrary map mapping an inputx∈ {0,1}nintosmemory bits,

andmquery algorithms running in timet. Here thei-th query algorithm answers queryiwhich is the

i-th output bit of f. Time is measured only by the number of queries that are made to the data structure, disregarding the processing time of such queries. The state of timetlower bounds for a given spacescan be summarized with the following expression:

t≥log(m/n)/log(s/n). (1.1) Specifically, no explicit function for which a bound better than (1.1) is known, for any setting of parameters. This is true even if time is measured in terms ofbitprobes (that is, the word size is 1), and the probes are non-adaptive. The latter means that the locations of the bit probes of thei-th query algorithm only depend oni.

Note that in such a data structure a query is simply answered by readingtbits from thesmemory bits at fixed locations that depend only on the query but not on the data. All the data structures in this note will be of this simple form, making our results stronger.

On the other hand, for several settings of parameters we can prove lower bounds that either match or are close to (1.1) for explicit functions. Specifically, forsuccinctdata structures using spaces=n+r

whenr=o(n), the expression (1.1) becomes(n/r)log(m/n). Gál and Miltersen [12], Theorem 5, have proved lower bounds of the formΩ(n/r).

Whens=n(1+Ω(1)), (1.1) is at best logarithmic. Such logarithmic lower bounds were obtained form=n1+Ω(1)by Siegel [23], Theorem 3.1, for computing hash functions. For several settings of parameters [23] also shows that the bound is tight. The lower bound was rediscovered in [15]. Their bounds are stated for non-binary, adaptive queries. For a streamlined exposition of this lower bound and matching upper bound, in the case of non-adaptive, binary queries see Lecture 18 in [26]. Remarkably, if

s=n(1+Ω(1))andm=O(s), or ifs=n1+Ω(1)no lower bound is known. Counting arguments (Theorem

8 in [17]) show the existence of functions requiring polynomial time even for spaces=m−1.

Miltersen, Nisan, Safra, and Wigderson [18] showed that data-structure lower bounds imply lower bounds for branching programs which read each variable few times. However stronger branching-program lower bounds were later proved by Ajtai [1] (see also [3]).

In this note we show that when mis close to s, say m=O(s) or m=s·poly logs even a slight improvement over the state of data structure lower bounds implies new circuit lower bounds. When

m=s1+ε for a small enoughε our connection still applies, but one would need to prove polynomial

lower bounds to obtain new circuit lower bounds. When saym=s2we do not obtain anything. It is an interesting open question to link that setting to circuit lower bounds.

2

Circuits with arbitrary gates

Acircuit C:{0,1}n→ {0,1}mwith arbitrary gatesis a circuit made of gates with unbounded fan-in,

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the depthd. These circuits have been extensively studied, see for example Chapter 13 in the book [14], titled “Circuits with arbitrary gates.” The best available lower bounds are proved in [22,8,7]. In the case of depth 2 they are polynomial, but for higher depths they are barely super-linear.

Definition 2.1. The functionλd:N→Nis defined asλ1(n):=b

nc,λ2(n):=dlog2neand ford≥3 λd(n):=λd?−2(n), where f?(n)is the least number of times we need to iterate the function f on inputn to reach a value≤1. Noteλ3(n) =O(log logn)andλ4(n) =O(log?n).

With this notation in hand we can express the best available lower bounds from [22,8,7]. They show explicit functions f:{0,1}n→ {0,1}mfor which any depth-dcircuit withwwires satisfies

w≥Ωd(m·λd−1(n)). (2.1) Those papers only consider the setting m=n, but we note that no lower bound better than (2.1) is available for m>n, because such a lower bound would immediately imply a bound better than

Ωd(n·λd−1(n))for some subset ofnoutput bits. (Write f :{0,1}n→ {0,1}mas f= (f1,f2, . . . ,fm/n)

where each fi:{0,1}n→ {0,1}n. If every fi is computable in depthd withw wires then f can be

computed in depthdwith(m/n)wwires. To make the reduction explicit we can append the indexito the input.)

Problem 2.2. Exhibit an explicit function f:{0,1}n→ {0,1}mthat cannot be computed by circuits of

depthdwithO(mλd−1(n))wires, with unbounded fan-in arbitrary gates, for somed.

We show how to simulate circuits with data structures. While it is natural to view a data structure as a depth-2 circuit, a key feature of our result is that it handles depth-dcircuits for anyd.

Theorem 2.3. Suppose the function f:{0,1}n→ {0,1}mhas a circuit of depthdwithwwires, consisting

of unbounded fan-in, arbitrary gates. Then f has a data structure with spaces=n+rand time(w/r)d, for anyr.

Proof. LetRbe a set ofrgates with largest fan-in in the circuit. NoteRmay include some of the output gates, but does not include any of the input. Note that any other gate has fan-in≤w/r, for else there arer

gates with fan-in>w/r, for a total of>wwires. The data structure simply stores in memory the input and the values of the gates inR. This takes spaces=n+r. It remains to show how to answer queries efficiently.

Group the gates of the circuit ind+1 levels where level 0 contains then input gates and leveld

contains the outputs. We prove by induction onithat for everyi=0,1, . . . ,d, a gate at levelican be computed by reading(w/r)ibits of the data structure. Fori=dthis gives the desired bound.

The base casei=0 holds as every input gate is stored in the data structure and thus can be computed by reading(w/r)0=1 bit.

Fixi>0 and a gateg. Ifg∈Rthen again it can be computed by reading 1 bit. Otherwiseg6∈R. Then

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This theorem shows that if for an explicit function f:{0,1}n→ {0,1}mwe have a data-structure

lower bound showing that for spaces=n+rthe timetmust be

t≥(ω(mλd−1(n))/r)d, (2.2) for somed, then we have new circuit lower bounds andProblem 2.2is solved. We illustrate this via several settings of parameters.

Setting s=1.01nandm=100n. As mentioned earlier in this setting no data structure lower bound is available: (1.1) gives nothing. We obtain that even proving, say, at≥log???(n)lower bound would solve Problem 2.2. Indeed, pickd=100 andr=0.01n. By Inequality (2.2) to solveProblem 2.2it suffices to prove a lower bound oft≥ω(λ99(n))100, which is implied byt≥log???(n).

The succinct setting s=n+rwithr=o(n),m=O(n). In this setting the best available lower bound ist≥Ω(n/r), see [12], Theorem 5. For sayd=5 andr≤n/log?n, the right-hand side of (2.2) is within a polynomial ofn/r. Hence for any setting ofrthe lower bound in [12] is within a polynomial of the best possible that one can obtain without solvingProblem 2.2. In particular, for redundancyr=n/logc(n), we have lower boundsΩ(logcn), and proving log5c(n)would solveProblem 2.2. We note that moreover

the data-structure given byTheorem 2.3issystematic: the input is copied innof then+rmemory bits. Thus the connection toProblem 2.2holds even for lower bounds for systematic data structures.

The setting m=n1+ε and s=n(1+Θ(1)). As mentioned earlier, the best known lower bound is

Ω(logm). We obtain that proving a data-structure lower bound of the formt≥n3εlog4nwould solve

Problem 2.2. (Pickr=nandd=3.)

The setting s=n1+Ω(1)and m=s1+ε.Here no data-structure lower bounds are known. We get that a

lower bound oft≥s3εlog4nwould solveProblem 2.2.

3

Bounded fan-in circuits

We also get a connection with bounded fan-in circuits (over the usual basis And, Or, Not). Recall that it is not known if every explicit function f:{0,1}n→ {0,1}mhas circuits of sizeO(m)and depth O(logm). Using Valiant’s well-known connection [24] (see [25, Chapter 3] for an exposition) we obtain the following.

Theorem 3.1. Let f :{0,1}n→ {0,1}mbe a function computable by bounded fan-in circuits withO(m)

wires and depthO(logm). Supposem=O(n). Then f has a data structure with spacen+o(n)and time

no(1).

Proof. Suppose the circuit hasw=cmwires and depthd=clogm. It is known [24] (see Lemma 28 in [25]) that we can halve the depth by removingcm/logdwires. Repeating this process say log log logn

times the depth becomesO(logn)/log logn=o(logn), and we have removedo(m)wires. LetRbe the set of wires we removed. The data structure consists of the input and the values ofR. Thus the redundancy is|R|=o(n).

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In the other uses of Valiant’s result, for depth-3 circuits and matrix rigidity, it is not important that each gate depends on few bits ofR. However it is essential for us. For this reason it is not clear if the corresponding depth-reduction for Valiant’s series-parallel circuits [6] yields data structures.

For completeness we discuss briefly lower bounds fordynamicdata structures. These data structures are not populated by an arbitrary map as in the static case above. Instead they start blank and are populated via additional insertion(or update) algorithms whose time is also taken into account. Here the best known lower bounds areΩ(log1.5n)[16]. We note that for an important class of problems, known as decomposableproblems, it is known since [4] how to turn a static data structure with query timetinto a data structure that supports queries in timeO(tlogn)as well as insertions in timeO(logn), see Theorem 7.3.2.5 in the book [20]. Hence a strong lower bound for such “half-dynamic” data structures would imply a lower bound for static data structures, and one can apply the above theorems to get a consequence for circuits.

4

Data structures for error-correcting codes

We can useTheorem 2.3to obtain new data structures for any problem which has efficient circuits. Let

f :{0,1}n→ {0,1}m be the encoding map of a binary error-correcting code which isasymptotically good, that ism=O(n)and the minimum distance isΩ(m). Gál and Miltersen show [12], Theorem 5,

that any data structure for this problem requires time≥Ω(n/r)if the space iss=n+r. Combining a slight extension ofTheorem 2.3together with a circuit construction in [11] we obtain data structures with timeO(n/r)log3n.

We first state the result from [11] that we need.

Theorem 4.1. [11] There exists a map f :{0,1}n→ {0,1}mwhich is the encoding map of an

asymp-totically good, binary error-correcting code and which is computable by a circuit with the following properties:

(1) The circuit has depth two and is made of XOR gates, (2) it hasO(nlog2n)wires,

(3) the fan-in of the output gates isO(logn), and (4) the fan-out of the input gates isO(log2n).

Remark 1. Actually the bounds in [11] are slightly stronger. (They prove that the minimum number of wires isΘ(n(logn/log logn)2).) We focus on the above parameters for simplicity. Properties (1)-(3) are

explicit in [11], Section 6. Property (4) is only needed here for the dynamic data structure in Theorem (4.3). It holds because the gates in the middle layer are grouped inO(logm)range detectors. And each range detector is constructed with a unique-neighbor expander where the degree in the input nodes is

O(logm).

Next is the new data structure.

Theorem 4.2. There exists an asymptotically good, binary code whose encoding map f :{0,1}n

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Proof. First note that if inTheorem 2.3we start with a circuit where the output gates have fan-ink, then we can have a data structure with timek(w/r)d−1. The proof is the same as before, using the bound ofk instead ofw/rfor the output gates. UsingTheorem 4.1the result follows.

We also obtain a dynamic data structure for the encoding map.

Theorem 4.3. There exists an asymptotically good, binary code whose encoding map f :{0,1}n

{0,1}m has dynamic data structure with space O(nlogn) supporting updating an input bit in time O(log2n), and computing one bit of the codeword in timeO(logn).

Proof. Suppose f has a depth-2 circuit where the output gates have fan-inkand the input gates have fan-out`. Then this gives a data structure where the memory consists of the middle layer of gates. To compute one bit of the codeword we simply read thekcorresponding bits in the data structure, and to update one message bit we simply update the`corresponding bits. Essentially this observation already appears in [5], except they do not parameterize it by the fan-in and fan-out, and so do not get a worst-case data structure. UsingTheorem 4.1the result follows.

In both data structures, the query algorithms are explicit, but the preprocessing and updates are not (because the corresponding layer in the circuits in [11] is not explicit).

5

A lower bound for large

s

As remarked earlier we have no lower bounds whensis much larger thann. Next we prove a lower bound oft≥3 for anym=nO(1)even fors=m/2−1. This also appeared in [26], Lecture 18. The lower bound is established for asmall-bias generator[19]. A function f:{0,1}n→ {0,1}mis an

ε-biased generator if the XOR of any subset of the output bits is equal to one with probabilitypsuch that|p−1/2| ≤ε, over a uniform input. There are explicit constructions withn=O(logm/ε)[19,2].

Theorem 5.1. Let f:{0,1}n→ {0,1}mbe ao(1)-biased generator. Suppose f has a data structure with

timet=2. Thens≥m/2.

Proof. Suppose for a contradiction thats<m/2. By inspection, any functiongon 2 bits is eitheraffine, or else isbiased, that is eitherg−1(1)org−1(0)contains at most one input.

Suppose≥m/2 of the queries are answered with affine functions. Then becauses<m/2 some linear combination of these affine queries is fixed. This is a contradiction.

Otherwise≥m/2 of the queries are answered with biased functions. We claim that there exists one biased query whose (set of two) probes are covered by the probes of other two biased queries. To show this we can keep collecting biased queries whose probes are not covered. We must stop eventually, for

s<m/2. Hence assume that the probes of f3are covered by those of f1and f2.

Because f1 and f2 are biased, for each of these two functions there exists an output value that determines the values of both its input bits. If these values occur for both f1and f2then the value of f3

is fixed as well. Hence, there exists an output combination of(f1,f2,f3)∈ {0,1}3which never occurs. This means that the distribution of(f1,f2,f3)over a uniform input has statistical distanceΩ(1)from

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see for example [13], the distribution of(f1,f2,f3)is

8·o(1) =o(1)close to uniform in statistical distance.

Problem 5.2. Exhibit an explicit function f:{0,1}n→ {0,1}mthat does not have a data structure with

spaces=n+m/10, and timet=3.

6

Comparison with independent work

Comparison with [10]. Independent work by Dvir, Golovnev, and Weinstein [10] connects data-structure lower bounds andmatrix rigidity. Both [10] and this paper aim to link data-structure lower bounds to other challenges in computational complexity. However the works are incomparable. [10] is concerned with linear data structures, and links them to rigidity or linear circuits. This paper is concerned with arbitrary data structures, and links them to circuits with arbitrary gates. [10] shows that a polylogarithmic data-structure lower bound, even against spaces=1.001nandm=n100would give new rigid matrices. This work gives nothing in this regime. The main results in [10] and this work are proved via different techniques. But each paper also has a result whose proof uses Valiant’s depth reduction [24].

Comparison with [9]. Independent work by Corrigan-Gibbs and Kogan [9] shows that better data-structure lower bounds for thefunction-inversionproblem (and related problems) yield new circuit lower bounds for depth-2 circuits with advice bits.

Acknowledgment. I am grateful to Omri Weinstein for stimulating discussions during his visit at Northeastern University and at the Simons institute. I also thank the anonymous referees for helpful comments.

References

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[26] EMANUELE VIOLA: Special topics in complexity theory, 2017. Available onauthor’s webpage. 2, 6

AUTHOR

Emanuele Viola Associate professor

Khoury College of Computer Sciences Northeastern University

Boston, MA viola ccs neu edu

http://www.ccs.neu.edu/~viola

ABOUT THE AUTHOR

EMANUELEVIOLA, after worrying about it for half his life, has decided that he believes

References

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